ANNALES SOCIETATIS M ATHEMATICAE POLON A E Series I: COM M ENTATIONES M ATHEMATICAE XXII (1980) R O CZN IKI PO LSK IEG O TOWARZYSTWA MATEM ATYCZNEGO
Séria I: PRACE M ATEMATYCZNE XXII (1980)
Ma r iu sz Ko r a s
(Warszawa)
Deformations of actions of linearly reductive groups
Let G be an algebraic group. By an algebraic family of actions of G on an algebraic variety X parametrized by an algebraic variety T we mean a family {(pt}teT of actions of G on X , such that the map (t , g>, x ) h> q>t{g, x) of T x G x X into X is regular. Two families {(pt}teT and {ф(}геТ~аге equivalent iff there exists an algebraic family {ht}teT of automorphisms X such that
<Pt(g, ht (x)) = КФг(д> x) for every g, t, x.
A family is trivial iff it is equivalent to the constant family. We say that the action <pt is a deformation of the action cp2 iff there exists an algebraic family of actions of G on X , parametrized by an connected variety, which contains (pl and q>2.
In the differential case, every differential family of actions of a compact Lie group on a compact manifold parametrized by [0, 1] is trivial (R. S. Palais and T. Stewart [3]).
In this note we shall show that if G is linearly reductive group, X is a complete variety, then every deformation of (p is equivalent to q>, and if t0e T is non-singular, then the family is trivial over an etale neighbour
hood of t0.
All considered algebraic varieties and their morphisms are defined over an algebraically closed field k.
It is known (H. Matsumura, F. Oort [2]), that for X complete, Aut X has a natural structure of an algebraic group (in general with infinitely many connected components).
Le m m a
1. Let G be an affine group and X a complete variety. Let {(pt}t€T be an algebraic family of actions of G on X parametrized by a connected variety T. Let ht: G -*■ Aut X denote the homomorphism induced by the action (pt. Then there exists an affine subgroup H a Aut X such that V ht (G) c= H,
t e T
P roof. There exists an affine, closed, normal, connected subgroup of Aut X which contains all connected affine subgroups of Aut X , [4]. T x G
6 — Prace Matematyczne 22.1
82 M. K o r a s
has finitely many irreducible components thus (J ht(G) has non-empty
t e T
intersection with only finitely many cosets of H 1 in Aut X . Then the union of such the cosets is an affine subgroup which contains all ht (G).
Le m m a
2. Let G be linearly reductive and let Y be an affine non-singular variety. Let T be a curve. Assume that G acts on T x Y in such a way that for every g the diagram
g: T x Y~-> T x Y
4 ^
T
commutes. Let x0 = (t0, y 0) be a fixed point o f the action. Then there exists an open neighbourhood U of t0 such that for t e U the induced action on {t} x Y has a fixed point.
P roof. We may assume that T is affine and non-singular by taking T'-normalization T and the induced action on V x Y. Let p: T x Y -> T be the projection. Ker dp\Xo is a G-invariant subspace of the tangent space TXQ( T x Y ) . There exists a G-invariant subspace W such that TXQ( T x Y )
= ker dp\x ® W. We can find a G-invariant irreducible closed subvariety X а: T x Y such that x 0 e X , TXQ(X) = W and x 0 is non-singular on X, cf. [1]. Since G acts trivially on W, G acts trivially on X. Since d(p\x)\X0:
W T t0(T) is an isomorphism, p\x is etale in some neighbourhood V of x0. Thus U = p(V) has the required property.
Th e o r e m.
Let G be an affine linearly reductive group. Let X , T and {<ptjteT be as in Lemma 1, let t0 e T. Then for every t e T the actions <ptQ and (pt are equivalent, i.e. there exists an automorphism h: X -> X satisfying h<pt0{g,x) = (pt (g,hx). I f T is non-singular at t0, then the family {<р,},бГ IS trivial over some etale neighbourhood o f t0.
P roof. Let ht: G -*■ Aut X be the homomorphism induced by q>t. By Lemma 1, there exists an affine subgroup Я <= Aut X such that V ü,(G) с Я.
t e T
We define an algebraic family {«At},6r of actions of G on Я by ipt( g , f )
— К (g~1) o f o h t (g). Let t xeT. Then there exists a connected curve V in T containing t0 and tx. By Lemma 2, there exists an open subset Ut T containing t0 such that, for every t e UtQ, \j/t has a fixed point.
This implies that the actions (ptQ and <pt for t e UtQ are equivalent. Similarly, for every t e T , there exists an open Ut c= T such that q>t and tpu for u e U t are equivalent. Thus (ptQ and (ptl are equivalent.
Assume now that T is non-singular at t0. Then there exists Y c T x H such that the projection Y -*■ T is an etale morphism in some neighbour
hood Y' of t0 and G acts trivially on Y. Thus, the induced family of actions
of G on Я over Y' has a section Y ' - * ( Y ' x H ) G. Therefore this family
is trivial.
Lineary reductive groups 83
Assumption about linear reductivity of G is essential. For example, let us consider the family {(pt}t A1 of actions of the additive group k + on P l , where <pt(g, {Уо, У1}) — {Уо + 9^1
уУ
1} ■ Then q>0 is the trivial action and is not equivalent to any q>t for t Ф 0.
Open Question.
Let {(pt}teT be an algebraic family of actions of G on an affine variety X = Spec A , where A is a finitely generated /c-algebra.
This family induces the family {ф(}{еТ of actions of G on A. Is it true, that for every a e A there exists a finite dimensional vector subspace in A, containing a and ф{-invariant for every tl
If the answer is positive, then one can prove the above theorem in the case where X is affine.
References
[1] A. B ia ly n ic k i-B ir u la , Some theorems on actions of algebraic groups, Ann. of Math. 98, 3 (1973).
[2] H. M a tsu m u ra , F. O o rt, Representability of group tiÿctors and automorphisms of algebraic schemes, Inventiones Math. 4 (1967/68).
[3] R. S. P a la is, T. S te w a r t, Deformations of compact differentiable transformation groups, Amer. J. Math. 82 (1961).
[4] M. R o s e n lic h t, Some basic theorems on algebraic groups, Amer. J. Math. 78(19“56).
